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Finite injury and Σ1-induction

  • Michael Mytilinaios (a1)

Abstract

Working in the language of first-order arithmetic we consider models of the base theory P. Suppose M is a model of P and let M satisfy induction for Σ1-formulas. First it is shown that the Friedberg-Muchnik finite injury argument can be performed inside M, and then, using a blocking method for the requirements, we prove that the Sacks splitting construction can be done in M. So, the “amount” of induction needed to perform the known finite injury priority arguments is Σ1-induction.

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Department of Mathematics, University of Crete, 71409 Iraklio, Greece

References

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[1]Friedberg, R. M., Two recursively enumerable sets of incomparable degrees of unsolvability, Proceedings of the National Academy of Sciences of the United States of America, vol. 43 (1957), pp. 236238.
[2]Kirby, L. A. S., Initial segments of models of arithmetic, Ph.D. thesis, University of Manchester, Manchester, 1977.
[3]Kreisel, G., Some reasons for generalizing recursion theory, Logic Colloquium '69 (Gandy, R. O. and Yates, C. E., editors), North-Holland, Amsterdam, 1971, pp. 139198.
[4]Muchnik, A. A., On the unsolvability of the problem of reducibility in the theory of algorithms, Doklady Akademii Nauk SSSR, vol. 108 (1956), pp. 194197. (Russian)
[5]Paris, J. B. and Kirby, L. A. S., Σ n-collection schemas in arithmetic, Logic Colloquium '77, North-Holland, Amsterdam, 1978, pp. 199209.
[6]Post, E. L., Recursively enumerable sets of integers and their decision problems, Bulletin of the American Mathematical Society, vol. 50 (1944), pp. 285316.
[7]Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.
[8]Sacks, G. E., On degrees less than 0′, Annals of Mathematics, ser. 2, vol. 77 (1963), pp. 211231.
[9]Sacks, G. E. and Simpson, S. G., The α-finite injury method, Annals of Mathematical Logic, vol. 4 (1972), pp. 343368.
[10]Shore, R. A., Splitting an α-recursively enumerable set, Transactions of the American Mathematical Society, vol. 204 (1975), pp. 6578.
[11]Simpson, S. G., private correspondence, 05 24, 1984. (Contained the proof of Theorem 3.2 and a discussion of the program.)

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Finite injury and Σ1-induction

  • Michael Mytilinaios (a1)

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